Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 26.3s

Precision: 64

Math

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

↓

\[\left(\cos th \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}\right) \cdot \left(a2 \cdot a2\right) + \frac{a1}{\sqrt[3]{\sqrt{2}}} \cdot \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right)\]

C

double f(double a1, double a2, double th) {
        double r1211144 = th;
        double r1211145 = cos(r1211144);
        double r1211146 = 2.0;
        double r1211147 = sqrt(r1211146);
        double r1211148 = r1211145 / r1211147;
        double r1211149 = a1;
        double r1211150 = r1211149 * r1211149;
        double r1211151 = r1211148 * r1211150;
        double r1211152 = a2;
        double r1211153 = r1211152 * r1211152;
        double r1211154 = r1211148 * r1211153;
        double r1211155 = r1211151 + r1211154;
        return r1211155;
}

↓

double f(double a1, double a2, double th) {
        double r1211156 = th;
        double r1211157 = cos(r1211156);
        double r1211158 = 0.5;
        double r1211159 = 2.0;
        double r1211160 = sqrt(r1211159);
        double r1211161 = r1211158 / r1211160;
        double r1211162 = cbrt(r1211161);
        double r1211163 = r1211157 * r1211162;
        double r1211164 = a2;
        double r1211165 = r1211164 * r1211164;
        double r1211166 = r1211163 * r1211165;
        double r1211167 = a1;
        double r1211168 = cbrt(r1211160);
        double r1211169 = r1211167 / r1211168;
        double r1211170 = r1211168 * r1211168;
        double r1211171 = r1211167 / r1211170;
        double r1211172 = r1211157 * r1211171;
        double r1211173 = r1211169 * r1211172;
        double r1211174 = r1211166 + r1211173;
        return r1211174;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

1. Initial program 0.5

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
2. Using strategy rm

3. Applied div-inv 0.5

   \[\leadsto \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2}}\right)} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

4. Applied associate-*l* 0.5

   \[\leadsto \color{blue}{\cos th \cdot \left(\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

5. Simplified 0.5

   \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
6. Using strategy rm

7. Applied add-cube-cbrt 0.5

   \[\leadsto \cos th \cdot \frac{a1 \cdot a1}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

8. Applied times-frac 0.5

   \[\leadsto \cos th \cdot \color{blue}{\left(\frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}}\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

9. Applied associate-*r* 0.5

   \[\leadsto \color{blue}{\left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
10. Using strategy rm

11. Applied add-cbrt-cube 0.8

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \frac{\cos th}{\color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

12. Applied add-cbrt-cube 0.8

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \frac{\color{blue}{\sqrt[3]{\left(\cos th \cdot \cos th\right) \cdot \cos th}}}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}} \cdot \left(a2 \cdot a2\right)\]

13. Applied cbrt-undiv 0.6

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \color{blue}{\sqrt[3]{\frac{\left(\cos th \cdot \cos th\right) \cdot \cos th}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

14. Simplified 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \sqrt[3]{\color{blue}{\frac{\frac{\cos th \cdot \left(\cos th \cdot \cos th\right)}{2}}{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]
15. Using strategy rm

16. Applied *-un-lft-identity 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \sqrt[3]{\frac{\frac{\cos th \cdot \left(\cos th \cdot \cos th\right)}{2}}{\sqrt{\color{blue}{1 \cdot 2}}}} \cdot \left(a2 \cdot a2\right)\]

17. Applied sqrt-prod 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \sqrt[3]{\frac{\frac{\cos th \cdot \left(\cos th \cdot \cos th\right)}{2}}{\color{blue}{\sqrt{1} \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

18. Applied div-inv 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \sqrt[3]{\frac{\color{blue}{\left(\cos th \cdot \left(\cos th \cdot \cos th\right)\right) \cdot \frac{1}{2}}}{\sqrt{1} \cdot \sqrt{2}}} \cdot \left(a2 \cdot a2\right)\]

19. Applied times-frac 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \sqrt[3]{\color{blue}{\frac{\cos th \cdot \left(\cos th \cdot \cos th\right)}{\sqrt{1}} \cdot \frac{\frac{1}{2}}{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

20. Applied cbrt-prod 0.5

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \color{blue}{\left(\sqrt[3]{\frac{\cos th \cdot \left(\cos th \cdot \cos th\right)}{\sqrt{1}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}\right)} \cdot \left(a2 \cdot a2\right)\]

21. Simplified 0.4

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \left(\color{blue}{\cos th} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}\right) \cdot \left(a2 \cdot a2\right)\]

22. Simplified 0.4

    \[\leadsto \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \frac{a1}{\sqrt[3]{\sqrt{2}}} + \left(\cos th \cdot \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]

23. Final simplification 0.4

    \[\leadsto \left(\cos th \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}\right) \cdot \left(a2 \cdot a2\right) + \frac{a1}{\sqrt[3]{\sqrt{2}}} \cdot \left(\cos th \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right)\]

Reproduce

herbie shell --seed 2019155 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))